Hi murt and welcome,
1. Yes you can get a virus simply by being conected to the internet. Do you have a firewall? MS Blaster being the most infamous one.
2. I would say that it would be normal if you didn't have protection, but with protection you are at least geting warnings.
You don't say what your Operating System is, most probably windows of some form. You should be visiting the windows update site at least once a month to keep upto date with patches/fixes, etc.
There a huge number of known vulnerabilities in windows and unless you patch these vulnerabilities (critical updates) your computer is vulnerable.
You need to let us know some more about your problems with your email in order for us to help, such as.
What email program you are using - the error mesages you are getting when you try to use it, etc.
But your first port of call should be the avast help file by clicking windows Start button>All Programs>Avast>Help, if its not their them it can be found in windows explorer
"c:\Program Files\Alwil Software\Avast4\ENGLISH\HELP\help.chm" that is assuming you have the english version otherwise the \ENGLISH\ needs to be changed to your language.
HTH David